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Chapter 5 The Time Value of Money

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Learning Objectives Explain the mechanics of compounding, and bringing the value of money back to the present. Understand annuities. Determine the future or present value of a sum when there are nonannual compounding periods. Determine the present value of an uneven stream of payments and understand perpetuities.

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**COMPOUND INTEREST, FUTURE, AND PRESENT VALUE**

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**Using Timelines to Visualize Cash Flows**

Timeline of cash flows Negative: Cash outflow; Positive: cash inflow So out 100 now, in 30 one year later; in 20 two year later; out 10 three year later; in 50 four year later.

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Compound Interest Compounding is when interest paid on an investment during the first period is added to the principal; then, during the second period, interest is earned on the new sum (that includes the principal and interest earned so far).

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Compound Interest Example: Compute compound interest on $100 invested at 6% for three years with annual compounding. 1st year interest is $6.00 Principal now is $106.00 2nd year interest is $6.36 Principal now is $112.36 3rd year interest is $6.74 Principal now is $119.10 Total interest earned: $19.10

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Future Value Future Value is the amount a sum will grow to in a certain number of years when compounded at a specific rate. FVN = PV (1 + r)n FVN = the future of the investment at the end of “n” years r = the annual interest (or discount) rate n = number of years PV = the present value, or original amount invested at the beginning of the first year

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Future Value Example Example: What will be the FV of $100 in 2 years at interest rate of 6%? FV2 = PV(1 + r)2 = $100 ( )2 = $100 (1.06)2 = $112.36

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**How to Increase the Future Value?**

Future Value can be increased by: Increasing number of years of compounding (N) Increasing the interest or discount rate (r) Increasing the original investment (PV) See example on next slide

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Changing R, N, and PV a. You deposit $500 in bank for 2 years. What is the FV at 2%? What is the FV if you change interest rate to 6%? FV at 2% = 500*(1.02)2 = $520.2 FV at 6% = 500*(1.06)2 = $561.8 b. Continue the same example but change time to 10 years. What is the FV now? FV = 500*(1.06)10= $895.42 c. Continue the same example but change contribution to $1500. What is the FV now? FV = 1,500*(1.06)10 = $2,686.27

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Figure 5-1

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Figure 5-2

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**Figure 5-2 Figure 5-2 illustrates that we can increase the FV by:**

Increasing the number of years for which money is invested; and/or Investing at a higher interest rate.

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**Computing Future Values using Calculator or Excel**

Review discussion in the text book Excel Function for FV: = FV(rate,nper,pmt,pv)

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Present Value Present value reflects the current value of a future payment or receipt.

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**Present Value PV = FVn {1/(1 + r)n}**

FVn = the future value of the investment at the end of n years n = number of years until payment is received r = the interest rate PV = the present value of the future sum of money

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PV example What will be the present value of $500 to be received 10 years from today if the discount rate is 6%? PV = $500 {1/(1+0.06)10} = $500 (1/1.791) = $500 (0.558) = $279

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Figure 5-3

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**Figure 5-3 Figure 5-3 illustrates that PV is lower if:**

Time period is longer; and/or Interest rate is higher.

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Using Excel Excel Function for PV: = PV(rate,nper,pmt,fv)

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ANNUITIES

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Annuity An annuity is a series of equal dollar payments for a specified number of years. Ordinary annuity payments occur at the end of each period.

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**FV of Annuity Compound Annuity**

Depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow.

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FV Annuity - Example What will be the FV of a 5-year, $500 annuity compounded at 6%? FV5 = $500 ( )4 + $500 ( )3 + $500( )2 + $500 ( ) + $500 = $500 (1.262) + $500 (1.191) + $500 (1.124) + $500 (1.090) + $500 = $ $ $ $ $500 = $2,818.50

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Table 5-1

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**FV of an Annuity – Using the Mathematical Formulas**

FVn = PMT {(1 + r)n – 1/r} FV n = the future of an annuity at the end of the nth year PMT = the annuity payment deposited or received at the end of each year r = the annual interest (or discount) rate n = the number of years

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**FV of an Annuity – Using the Mathematical Formulas**

What will $500 deposited in the bank every year for 5 years at 10% be worth? FV = PMT ([(1 + r)n – 1]/r) = $500 (5.637) = $2,818.50

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**FV of Annuity: Changing PMT, N, and r**

What will $5,000 deposited annually for 50 years be worth at 7%? FV = $2,032,644 Contribution = 250,000 (= 5000*50) Change PMT = $6,000 for 50 years at 7% FV = 2,439,173 Contribution= $300,000 (= 6000*50)

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**FV of Annuity: Changing PMT, N, and r**

3. Change time = 60 years, $6,000 at 7% FV = $4,881,122 Contribution = 360,000 (= 6000*60) 4. Change r = 9%, 60 years, $6,000 FV = $11,668,753 Contribution = $360,000 (= 6000*60)

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**Present Value of an Annuity**

Pensions, insurance obligations, and interest owed on bonds are all annuities. To compare these three types of investments we need to know the present value (PV) of each.

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Table 5-2

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**PV of Annuity – Using the Mathematical Formulas**

PV of Annuity = PMT {[1 – (1 + r)–1]}/r = 500 (4.212) = $2,106

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Annuities Due Annuities due are ordinary annuities in which all payments have been shifted forward by one time period. Thus, with annuity due, each annuity payment occurs at the beginning of the period rather than at the end of the period.

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Annuities Due Continuing the same example: If we assume that $500 invested every year at 6% to be annuity due, the future value will increase due to compounding for one additional year. FV5 (annuity due) = PMT {[(1 + r)n – 1]/r} (1 + r) = 500(5.637)(1.06) = $2,987.61 (versus $2, for ordinary annuity)

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**MAKING INTEREST RATES COMPARABLE**

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**Making Interest Rates Comparable**

We cannot compare rates with different compounding periods. For example, 5% compounded annually is not the same as 5% percent compounded quarterly. To make the rates comparable, we compute the annual percentage yield (APY) or effective annual rate (EAR).

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**Quoted Rate versus Effective Rate**

Quoted rate could be very different from the effective rate if compounding is not done annually. Example: $1 invested at 1% per month will grow to $ (= $1.00(1.01)12) in one year. Thus even though the interest rate may be quoted as 12% compounded monthly, the effective annual rate or APY is 12.68%.

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**Quoted Rate versus Effective Rate**

EAR = (1 + quoted rate/m)m – 1 Where m = number of compounding periods = ( /12)12 – 1 = (1.01)12 – 1 = or %

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Table 5-4

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**Finding PV and FV with Nonannual Periods**

If interest is not paid annually, we need to change the interest rate and time period to reflect the nonannual periods while computing PV and FV. r = stated rate/# of compounding periods N = # of years * # of compounding periods in a year Example: If your investment earns 10% a year, with quarterly compounding for 10 years, what should we use for “r” and “N”? r = 0.10/4 = or 2.5% N = 10*4 = 40 periods

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**THE PRESENT VALUE OF AN UNEVEN STREAM AND PERPETUITIES**

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**The Present Value of an Uneven Stream**

Some cash flow stream may not follow a conventional pattern. For example, the cash flows may be erratic (with some positive cash flows and some negative cash flows) or cash flows may be a combination of single cash flows and annuity (as illustrated in Table 5-5).

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Table 5-5

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Table 5-7

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**Key Terms Effective annual rate Amortized loans Future value Annuity**

Future value factor Ordinary annuity Present value Present value factor Perpetuity Simple interest Amortized loans Annuity Annuity due Annuity future value factor Annuity present value factor Compound annuity Compound interest

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Table 5-6

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© 2009 Cengage Learning/South-Western The Time Value Of Money Chapter 3.

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